# I The cases in proving that group of order 90 is not simple

1. Dec 8, 2018 at 4:55 PM

### Mr Davis 97

https://imgur.com/a/FuCPJLe

I am trying to attempt this problem, but I am wondering why exactly these are the two cases the problem is split into. I can understand the first case, since that lets us count elements and get a contradiction, but why is the second case there? In other words, why do these two cases exhaust all possibilities?

EDIT: Actually, I think that I see it now. Since the intersection of subgroups is a subgroup, by Lagrange we must have that the negation of two distinct Sylow 3-subgroups intersecting trivially is intersecting with 3 elements.

Last edited: Dec 8, 2018 at 5:17 PM
2. Dec 9, 2018 at 12:48 AM